On reducibility of ultrametric almost periodic linear representations. (English) Zbl 0818.46083
Let \(G\) be a group, let \(K\) be a complete non-Archimedean valued field. A function \(G\to K\) is called almost periodic if its (left) translates form a compactoid for the uniform topology. Here compactoidity is a ‘convexified’ notion of precompactness which becomes useful if \(K\) is not separable. The almost periodic functions on \(G\) form a Banach algebra \(AP(G\to K)\) over \(K\). It is also a (complete) Hopf algebra under the comultiplication \(f\mapsto \pi\circ \Delta(f)\) where \(\Delta: AP(G\to K)\to AP(G\times G\to K)\) is given by \((\Delta f)(s, t)= f(st)\) and where \(\pi\) is the canonical isomorphism
\[
AP(G\times G\to K)\to AP(G\to K)\widehat\otimes AP(G\to K),
\]
the co-unit \(f\mapsto f(s)\) and antipode \(\eta\) given by \(\eta(f)(s)= f(s^{-1})\).
By using Hopf algebra techniques the following theorem is proved: Let \(AP(G\to K)\) have an invariant mean (i.e. a shift invariant element of \(AP(G\to K)'\) sending the constant function 1 into \(1\in K\)). Then
(i) any topologically irreducible almost periodic linear representation is finite-dimensional
(ii) if \(U\) is an almost periodic representation of \(G\) into some Banach space \(E\) then \(E\) is the topological direct sum of irreducible \(U\)- invariant subspaces.
By applying this theorem to the left regular representation of \(G\) into \(AP(G\to K)\), a Peter-Weyl-like theorem is obtained.
By using Hopf algebra techniques the following theorem is proved: Let \(AP(G\to K)\) have an invariant mean (i.e. a shift invariant element of \(AP(G\to K)'\) sending the constant function 1 into \(1\in K\)). Then
(i) any topologically irreducible almost periodic linear representation is finite-dimensional
(ii) if \(U\) is an almost periodic representation of \(G\) into some Banach space \(E\) then \(E\) is the topological direct sum of irreducible \(U\)- invariant subspaces.
By applying this theorem to the left regular representation of \(G\) into \(AP(G\to K)\), a Peter-Weyl-like theorem is obtained.
Reviewer: W.Schikhof (Nijmegen)
MSC:
46S10 | Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis |
43A60 | Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions |
16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |
Keywords:
complete non-Archimedean valued field; almost periodic; precompactness; Banach algebra; Hopf algebra; comultiplication; co-unit; antipode; invariant mean; topologically irreducible almost periodic linear representation; left regular representation; Peter-Weyl-like theoremReferences:
[1] | DOI: 10.1016/0019-3577(90)90039-P · Zbl 0709.43006 · doi:10.1016/0019-3577(90)90039-P |
[2] | Schikhof, An approach to p-adic almost periodicity by means of compactoids (1988) |
[3] | Rooij, Non-archimedean functional analysis (1978) · Zbl 0396.46061 |
[4] | Serre, Publ. Math. pp 69– (1962) |
[5] | Gruson, Bull. Soc. Math. France 94 pp 67– (1966) |
[6] | Diarra, Ultrametric almost periodic linear representations in p-adic Functional Analysis pp 45– (1994) |
[7] | Amice, Les nombres p-adiques (1975) |
[8] | Rangan, p-adic Functionnal Analysis pp 141– (1991) |
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